Encyclopedia of Microtonal Music Theory

tina

[Joe Monzo]

A very small unit of interval measurement, advocated by Joe Monzo primarly because of its accuracy for use as logarithmic integer values to describe just-intonation intervals thru the 31-limit, first suggested by privately by George Secor in September 2004, then publicly by Gene Ward Smith on 22 April 2007. A few days later, Monzo independently found, by using an error calculation that weighted the percent error for prime-factors as 3:*10, 5:*6, 7:*3, 11:*2, and all the others up to 41 *1, that 8539-edo had a lower "score" than any other EDO with cardinality smaller than 19932-edo, giving excellent approximations to all primes in the 41-limit except 37. Dave Keenan and George Secor referred to it as "tina", a name which Monzo adopted.

A tina divides the octave into 8,539 equal parts. Because that number is prime, it does not offer any other divisibility, which is its chief disadvantage.

The tina is therefore calculated as the 8539th root of 2, or 2(1/8539), with a ratio of approximately 1:1.000081178. It is an irrational number.

The 12-edo semitone is exactly 711 7/12 (= 711.58,3...) tinas, a value which is nearly midway between two adjacent tinas. The unfortunate result of this is that there is rounding error when using tinas to represent 12-edo. In actual practice it is not really a problem because one tina is so small, but it does cause problems with the mathematics. Below is a table (arranged in descending order of pitch) showing the tina values for 12-edo based on the chain-of-5ths from -5...+6 generators:

Below is a table of tina values for common 31-limit JI interval sizes; tina values are given both in integer and floating-point form, expressly to point out how unnecessary it is to use the decimal places for most of the cases:

Of those, the only really important ones for most people will be the semicomma, kleisma, and magic-comma. All the rest of the JI intervals in the table (except one) have an error of 17 percent or less. This a remarkably low level of error for an integer logarithmic measurement unit.

Below is a table of the tina values for all of the commonly used intervals, in all of the standard keys, in some of the most important EDO meantones. The intervals are listed as a chain-of-5ths, in decreasing generator order, with the tonic of each key as the zeroth generator. 53-edo is also shown for comparison, as a representation of pythagorean tuning. The percentage errors for 12-, 55-, and 31-edo are quite low, those for 43-, 50-, and 19-edo not as good, and the error for 53-edo almost as bad as it can get, at 49% (i.e., the 5th of 53-edo is almost midway between two tina values, at ~4994.5094) -- the values shown in the 53-edo column are actually quite accurate for real pythagorean JI tuning.